|
%I #37 Mar 21 2018 10:53:40
%S 1,2,5,12,28,60,124,256,512,1024,2048,4096,8192,16384,32768,65536,
%T 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,
%U 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592
%N Clique number for the n-Keller graph.
%C a(n) <= 2^n.
%C a(7) = 124 was established by Debroni et al. (2011).
%C a(8) = 2^8 was established by Mackey (2002).
%C a(n) = 2^n for n >= 8 (see Jarnicki et al.).
%H Colin Barker, <a href="/A202604/b202604.txt">Table of n, a(n) for n = 1..1000</a>
%H J. Debroni, J. D. Eblen, M. A. Langston, P. Shor, W. Myrvold, D. Weerapurage, <a href="http://hdl.handle.net/1721.1/81184">A complete resolution of the Keller maximum clique problem</a>, Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms, pp. 129-135, 2011.
%H Witold Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, <a href="https://arxiv.org/abs/1606.07918">Properties, Proved and Conjectured, of Keller, Mycielski, and Queen Graphs</a>, arXiv preprint arXiv:1606.07918 [math.CO], 2016.
%H J. Mackey, <a href="http://dx.doi.org/10.1007/s00454-002-2801-9">A cube tiling of dimension eight with no facesharing</a>, Discrete & Computational Geometry 28 (2): 275-279, 2002.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CliqueNumber.html">Clique Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KellerGraph.html">Keller Graph</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%F G.f.: x*(1 + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 4*x^6 + 8*x^7) / (1 - 2*x). - _Colin Barker_, Oct 14 2017
%t Table[Piecewise[{{1, n == 1}, {2, n == 2}, {5, n == 3}, {2^n - 4, 4 <= n <= 7}}, 2^n], {n, 20}] (* _Eric W. Weisstein_, Mar 21 2018 *)
%t Join[{1, 2, 5, 12, 28, 60, 124}, LinearRecurrence[{2}, {256}, 14]] (* _Eric W. Weisstein_, Mar 21 2018 *)
%t CoefficientList[Series[(-1 - x^2 - 2 x^3 - 4 x^4 - 4 x^5 - 4 x^6 - 8 x^7)/(-1 + 2 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 21 2018 *)
%o (PARI) Vec(x*(1 + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 4*x^6 + 8*x^7) / (1 - 2*x) + O(x^40)) \\ _Colin Barker_, Oct 14 2017
%Y Cf. A295902, A296100, A296101.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_, Dec 21 2011
%E More terms from _N. J. A. Sloane_, Jul 04 2017 based on the Jarnicki et al. survey.
|
